more search methods
DESCRIPTION
More Search Methods. Local Search Hill Climbing Simulated Annealing Beam Search Genetic Search Local Search in Continuous Spaces Searching with Nondeterministic Actions Online Search (agent is executing actions). Local Search Algorithms and Optimization Problems. - PowerPoint PPT PresentationTRANSCRIPT
More Search Methods
• Local Search– Hill Climbing– Simulated Annealing– Beam Search– Genetic Search
• Local Search in Continuous Spaces
• Searching with Nondeterministic Actions
• Online Search (agent is executing actions)
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Local Search Algorithms and Optimization Problems
• Complete state formulation– For example, for the 8 queens problem, all 8 queens
are on the board and need to be moved around to get to a goal state
• Equivalent to optimization problems often found in science and engineering
• Start somewhere and try to get to the solution from there
• Local search around the current state to decide where to go next
Pose Estimation Example
• Given a geometric model of a 3D object and a 2D image of the object.
• Determine the position and orientation of the object wrt the camera that snapped the image.
image 3D object
• State (x, y, z, x-angle, y-angle, z-angle) 4
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Hill Climbing “Gradient ascent”
solution
Note: solutions shownhere as max not min.
Often used for numerical optimization problems.
How does it work?
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AI Hill Climbing
Steepest-Ascent Hill Climbing• current start node• loop do
– neighbor a highest-valued successor of current– if neighbor.Value <= current.Value then return current.State– current neighbor
• end loop
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Hill Climbing Search
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4 10 3 2 8
current
What if current had a value of 12?
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Hill Climbing Problems
Local maxima
Plateaus
Diagonal ridges
What is it sensitive to?Does it have any advantages?
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Solving the Problems
• Allow backtracking (What happens to complexity?)
• Stochastic hill climbing: choose at random from uphill moves, using steepness for a probability
• Random restarts: “If at first you don’t succeed, try, try again.”
• Several moves in each of several directions, then test
• Jump to a different part of the search space
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Simulated Annealing
• Variant of hill climbing (so up is good)
• Tries to explore enough of the search space early on, so that the final solution is less sensitive to the start state
• May make some downhill moves before finding a good way to move uphill.
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Simulated Annealing
• Comes from the physical process of annealing in which substances are raised to high energy levels (melted) and then cooled to solid state.
• The probability of moving to a higher energy state, instead of lower is
p = e^(-E/kT) where E is the positive change in energy level, T is the temperature, and k is Bolzmann’s constant.
heat cool
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Simulated Annealing
• At the beginning, the temperature is high.• As the temperature becomes lower
– kT becomes lower– E/kT gets bigger– (-E/kT) gets smaller– e^(-E/kT) gets smaller
• As the process continues, the probability of a downhill move gets smaller and smaller.
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For Simulated Annealing
E represents the change in the value of the objective function.
• Since the physical relationships no longer apply, drop k. So p = e^(-E/T)
• We need an annealing schedule, which is a sequence of values of T: T0, T1, T2, ...
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Simulated Annealing Algorithm
• current start node;
• for each T on the schedule /* need a schedule */
– next randomly selected successor of current– evaluate next; it it’s a goal, return it
E next.Value – current.Value /* already negated */
– if E > 0
• then current next /* better than current */
• else current next with probability e^(E/T)
How would you do this probabilistic selection?
Probabilistic Selection
• Select next with probability p
• Generate a random number
• If it’s <= p, select next
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0 1prandom number
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Simulated Annealing Properties
• At a fixed “temperature” T, state occupation probability reaches the Boltzman distribution
• If T is decreased slowly enough (very slowly), the procedure will reach the best state.
• Slowly enough has proven too slow for some researchers who have developed alternate schedules.
p(x) = e^(E(x)/kT)
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Simulated Annealing Schedules
• Acceptance criterion and cooling schedule
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Simulated Annealing Applications
• Basic Problems– Traveling salesman– Graph partitioning– Matching problems– Graph coloring– Scheduling
• Engineering– VLSI design
• Placement• Routing• Array logic minimization• Layout
– Facilities layout– Image processing– Code design in information theory
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Local Beam Search
• Keeps more previous states in memory– Simulated annealing just kept one previous state in
memory.– This search keeps k states in memory.
- randomly generate k initial states
- if any state is a goal, terminate
- else, generate all successors and select best k
- repeat
Pros? Cons?
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Genetic Algorithms • Start with random population of states
– Representation serialized (ie. strings of characters or bits)
– States are ranked with “fitness function”
• Produce new generation
– Select random pair(s) using probability:
• probability ~ fitness
– Randomly choose “crossover point”
• Offspring mix halves
– Randomly mutate bits
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Crossover Mutation
Genetic Algorithm
• Given: population P and fitness-function f• repeat
– newP empty set– for i = 1 to size(P)
x RandomSelection(P,f)
y RandomSelection(P,f)
child Reproduce(x,y)
if (small random probability) then child Mutate(child)
add child to newP– P newP
• until some individual is fit enough or enough time has elapsed• return the best individual in P according to f
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Using Genetic Algorithms
• 2 important aspects to using them– 1. How to encode your real-life problem– 2. Choice of fitness function
• Research Example– We want to generate a new operator for
finding interesting points on images– The operator will be some function of a set of
values v1, v2, … vK.
– Idea: weighted sums, weighted mins, weighted maxs. a1,…aK,b1,…bK,c1,…ck
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Local Search in Continuous Spaces
• Given a continuous state space
S = {(x1,x2,…,xN) | xi R}
• Given a continuous objective function f(x1,x2,…,xN)
• The gradient of the objective function is a vector f = (f/x1,f/x2,…,f/xN)
• The gradient gives the magnitude and direction of the steepest slope at a point.
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Local Search in Continuous Spaces
• To find a maximum, the basic idea is to set f =0
• Then updating of the current state becomes
x x + f(x)
where is a small constant.
• Theory behind this is taught in numerical methods classes.
• Your book suggests the Newton-Raphson method. Luckily there are packages…..
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Computer Vision Pose Estimation Example
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Computer Vision Pose Estimation Example
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Read the “Airports in Romania” example in the text.
Searching with Nondeterministic Actions
• Vacuum World (actions = {left, right, suck})
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Searching with Nondeterministic Actions
In the nondeterministic case, the result of an action can vary.
Erratic Vacuum World: • When sucking a dirty square, it cleans it and
sometimes cleans up dirt in an adjacent square.
• When sucking a clean square, it sometimes deposits dirt on the carpet.
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Generalization of State-Space Model
1. Generalize the transition function to return a set of possible outcomes.
oldf: S x A -> S newf: S x A -> 2S
2. Generalize the solution to a contingency plan.
if state=s then action-set-1 else action-set-2
3. Generalize the search tree to an AND-OR tree.
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AND-OR Search Tree
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ANDNode
ORNode
Searching with Partial Observations
• The agent does not always know its state!
• Instead, it maintains a belief state: a set of possible states it might be in.
• Example: a robot can be used to build a map of a hostile environment. It will have sensors that allow it to “see” the world.
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Belief State Space for Sensorless Agent
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initial state
Knows it’s on the right.
Knows it’son the left
Knows leftside clean
Online Search Problems
• Active agent– executes actions– acquires percepts from sensors– deterministic and fully observable– has to perform an action to know the
outcome
• Examples– Web search– Autonomous vehicle
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